Finite Group Databasebeta

← C9×#5166 #13143345

Subgroups — C9×#5166 #13143345

Order 1728.

404 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×SD16C211yesmaximal, characteristic
2Q8×C9×Dic3C211yesmaximal, characteristic, complemented
3C2×C9×#216C211yesmaximal, characteristic, complemented
4C9×#936C211yesmaximal, characteristic
5C9×#832C211yesmaximal, characteristic
6C9×#816C211yesmaximal, characteristic, complemented
7C9×Dic3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5166C311yesmaximal, characteristic
9C9×#464—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C3×D4×C9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×SD16—22no—
14C2×C3×C8×C9C2211yescharacteristic
15C9×#217—21no—
16C9×#217—21no—
17C9×#216—22no—
18C9×C4⋊Dic3—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C4×C9×Dic3—21no—
21C4×C9×Dic3C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23C2×C32×SD16C2×C311yescharacteristic
24C3×Q8×Dic3C2×C311yescharacteristic
25C2×C3×#216C2×C311yescharacteristic
26C3×#936C2×C311yescharacteristic
27C3×#832C2×C311yescharacteristic
28C3×#816C2×C311yescharacteristic
29C3×Dic3⋊C8C2×C311yescharacteristic
30C2×C9×Q16—31no—
31C2×C9×SD16—21no—
32C2×C9×SD16D311yescharacteristic
33C9×#124—31no—
34C4×Q8×C9—31no—
35C9×C4⋊C8—31no—
36C9×#103—31no—
37C9×#102—31no—
38C23×C3×C9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9D412yes—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C9×Dic3D411yescharacteristic
48C2×C9×Dic3D411yescharacteristic
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×C8×C9—21no—
52C9×C3⋊C8—21no—
53C3×#464—31no—
54#5166C911yescharacteristic, direct product factor
55C2×C32×Q8C22×C311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic
57C2×C3×Dic6C22×C311yescharacteristic
58C32×SD16—22no—
59C2×C32×C8C22×C311yescharacteristic
60C3×#217—21no—
61C3×#217—21no—
62C3×#216—22no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3C22×C311yescharacteristic
65C3×C4×Dic3—21no—
66C3×C4×Dic3C22×C311yescharacteristic
67C2×C3×C3⋊C8C22×C311yescharacteristic
68C2×Q8×C9—21no—
69C2×Q8×C9—31no—
70C2×Q8×C9C2×D311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9C2×D311yescharacteristic
73C9×Q16—62no—
74C9×SD16—24no—
75C9×SD16—22no—
76C2×C8×C9—21no—
77C2×C8×C9—31no—
78C2×C8×C9C2×D311yescharacteristic
79C9×C4⋊C4—31no—
80C9×C4⋊C4—61no—
81C9×C22⋊C4—61no—
82C9×C22⋊C4—61no—
83C42×C9—31no—
84C42×C9—61no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9—41no—
88C22×C3×C9C2×D411yescharacteristic
89C3×C4×C9—22no—
90C3×C4×C9C2×D411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C9×Dic3—21no—
93C9×Dic3—21no—
94C9×Dic3—41no—
95C9×Dic3—41no—
96C2×C3×Q16—31no—
97C2×C3×SD16—21no—
98C2×C3×SD16C2×C911yescharacteristic
99C2×C3×SD16C3×D311yescharacteristic
100C3×#124—31no—
101C3×C4×Q8—31no—
102Q8×Dic3C2×C911yescharacteristic, complemented
103C2×#216C2×C911yescharacteristic, complemented
104#936C2×C911yescharacteristic
105C3×C4⋊C8—31no—
106C3×#103—31no—
107C3×#102—31no—
108#832C2×C911yescharacteristic
109#816C2×C911yescharacteristic, complemented
110Dic3⋊C8C2×C911yescharacteristic, complemented
111C23×C32—21no—
112C32×Q8—21no—
113C32×Q8C3×D412yes—
114C32×D4—21no—
115C32×D4—21no—
116C2×C32×C4—21no—
117C2×C32×C4C23×C311yescharacteristic
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3C3×D411yescharacteristic
121C2×C3×Dic3C3×D411yescharacteristic
122C3×Dic6—21no—
123C3×Dic6—21no—
124C23×C9—21no—
125C23×C9—41no—
126C32×C8—21no—
127C3×C3⋊C8—21no—
128Q8×C9—21no—
129Q8×C9—22no—
130Q8×C9—41no—
131Q8×C9—61no—
132Q8×C9—61no—
133Q8×C9#6712yes—
134D4×C9—22no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—41no—
138C2×C4×C9—21no—
139C2×C4×C9—21no—
140C2×C4×C9—31no—
141C2×C4×C9—31no—
142C2×C4×C9—41no—
143C2×C4×C9—61no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C8×C9—22no—
147C8×C9—21no—
148C8×C9—61no—
149#464—31noSylow
150C2×C3×C9—81no—
151C2×C3×C9#12111yescharacteristic
152C2×C3×C9#13511yescharacteristic
153C2×C3×C9#13711yescharacteristic
154C2×C3×Q8—21no—
155C2×C3×Q8—31no—
156C2×C3×Q8C22×C911yescharacteristic
157C2×C3×Q8C2×C3×D311yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4C22×C911yescharacteristic
160C2×C3×D4C2×C3×D311yescharacteristic
161C2×Dic6C22×C911yescharacteristic
162C3×Q16—62no—
163C3×SD16—24no—
164C3×SD16—22no—
165C3×SD16—22no—
166C2×C3×C8—21no—
167C2×C3×C8—31no—
168C2×C3×C8C22×C911yescharacteristic
169C2×C3×C8C2×C3×D311yescharacteristic
170C3×C4⋊C4—31no—
171C3×C4⋊C4—61no—
172C3×C22⋊C4—61no—
173C3×C22⋊C4—61no—
174C3×C42—31no—
175C3×C42—61no—
176#217—21no—
177#217—21no—
178#216—22no—
179C4⋊Dic3—21no—
180C4⋊Dic3C22×C911yescharacteristic
181C4×Dic3—21no—
182C4×Dic3C22×C911yescharacteristic
183C2×C3⋊C8C22×C911yescharacteristic
184C22×C32—41no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32C2×C3×D411yescharacteristic
188C32×C4—22no—
189C32×C4C2×C3×D411yescharacteristic
190C32×C4C3×#4111yescharacteristic
191C3×Dic3—21no—
192C3×Dic3—21no—
193C3×Dic3—41no—
194C3×Dic3—41no—
195C22×C9—21no—
196C22×C9—42no—
197C22×C9—41no—
198C22×C9—41no—
199C22×C9—41no—
200C22×C9—81no—
201C22×C9—81no—
202C22×C9D3×D411yescharacteristic
203C4×C9—22no—
204C4×C9—21no—
205C4×C9—21no—
206C4×C9—42no—
207C4×C9—61no—
208C4×C9—61no—
209C4×C9—121no—
210C4×C9—121no—
211C4×C9#23711yescharacteristic
212C4×C9C2×#6711yescharacteristic
213C2×Q16—31no—
214C2×SD16D3×C911yescharacteristic
215#124—31no—
216C4×Q8—31no—
217C4⋊C8—31no—
218#103—31no—
219#102—31no—
220C3×C9#46411yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—41no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—22no—
227C3×Q8—41no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8D4×C912yes—
231C3×Q8C3×#6712yes—
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—41no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—41no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4C23×C911yescharacteristic
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3D4×C911yescharacteristic
251C2×Dic3D4×C911yescharacteristic
252Dic6—21no—
253Dic6—21no—
254C3×C8—22no—
255C3×C8—21no—
256C3×C8—21no—
257C3×C8—61no—
258C3⋊C8—21no—
259C2×C32—81no—
260C2×C32C3×#12111yescharacteristic
261C2×C32C3×#13511yescharacteristic
262C2×C32C3×#13711yescharacteristic
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—82no—
267C2×C9—81no—
268C2×C9#91511yescharacteristic
269C2×C9#91611yescharacteristic
270C2×C9#93911yescharacteristic
271C2×Q8—31no—
272C2×Q8C2×D3×C911yescharacteristic
273C2×D4C2×D3×C911yescharacteristic
274Q16—62no—
275SD16—22no—
276C2×C8—31no—
277C2×C8C2×D3×C911yescharacteristic
278C4⋊C4—31no—
279C4⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—31no—
283C42—61no—
284C22×C3—21no—
285C22×C3—42no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—81no—
293C22×C3—81no—
294C22×C3C2×D4×C911yescharacteristic
295C22×C3C3×D3×D411yescharacteristic
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—42no—
301C3×C4—61no—
302C3×C4—61no—
303C3×C4—121no—
304C3×C4—121no—
305C3×C4C2×D4×C911yescharacteristic
306C3×C4C9×#4111yescharacteristic
307C3×C4C3×#23711yescharacteristic
308C3×C4C2×C3×#6711yescharacteristic
309Dic3—21no—
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313C32C3×#46411yescharacteristic
314C9—21no—
315C9#516611yescharacteristic, direct product factor
316C23—21no—
317Q8—21no—
318Q8—61no—
319Q8—61no—
320Q8C9×#6712yes—
321D4—21no—
322D4—21no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4C22×D3×C911yescharacteristic
329C8—21no—
330C8—61no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—82no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3C9×#12111yescharacteristic
338C2×C3C9×#13511yescharacteristic
339C2×C3C9×#13711yescharacteristic
340C2×C3C3×#91511yescharacteristic
341C2×C3C3×#91611yescharacteristic
342C2×C3C3×#93911yescharacteristic
343C22—41no—
344C22—41no—
345C22—41no—
346C22D3×D4×C911yescharacteristic
347C4—22no—
348C4—61no—
349C4—61no—
350C4—121no—
351C4—121no—
352C4C9×#23711yescharacteristic
353C4C2×C9×#6711yescharacteristic
354C3—21no—
355C3C9×#46411yescharacteristic, complemented
356C3C3×#516611yescharacteristic
357C2—81no—
358C2C9×#91511yescharacteristic
359C2C9×#91611yescharacteristic
360C2C9×#93911yescharacteristic
subgroupG/H
centerC22×C9D3×D4
commutatorC2×C3×C4C23×C9
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C3×C9×SD16C2
socleC22×C32C2×C3×D4
prime radicalC2×C32×D4C22×C3
cosocleC2×C32×C4C23×C3
hypercenterC2×C9×SD16D3
hypocenterC3C9×#464
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C9 ◁ C2×C4×C9 ◁ C2×C9×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. Q8×C9×Dic3C2semidirect
3. C2×C9×#216C2semidirect
6. C9×#816C2semidirect
7. C9×Dic3⋊C8C2semidirect
54. #5166C9direct (N × H)
102. Q8×Dic3C2×C9semidirect
103. C2×#216C2×C9semidirect
109. #816C2×C9semidirect
110. Dic3⋊C8C2×C9semidirect
220. C3×C9#464semidirect
315. C9#5166direct (N × H)
355. C3C9×#464semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×C9×SD162
Q8×C9×Dic32
C2×C9×#2162
C9×#9362
C9×#8322
C9×#8162
C9×Dic3⋊C82
C3×#51663
C9×#4643